Journal of Algebra 423 (2015) 405–421 Contents lists available at ScienceDirect Journal of Algebra www.elsevier.com/locate/jalgebra Generic extensions and generic polynomials for multiplicative groups Jorge Morales a, Anthony Sanchez b,1 a Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA b Department of Mathematics, Arizona State University, Tempe, AZ 85281, USA a r t i c l e i n f o a b s t r a c t Article history: Let A be a finite-dimensional algebra over a finite field Fq × Received 23 June 2013 and let G = A be the multiplicative group of A. In this Available online 31 October 2014 paper, we construct explicitly a generic Galois G-extension Communicated by Eva S/R, where R is a localized polynomial ring over Fq, and an Bayer-Fluckiger A explicit generic polynomial for G in dimFq ( ) parameters. MSC: © 2014 Published by Elsevier Inc. 12F12 13B05 Keywords: Constructive Galois theory Frobenius modules Generic polynomials Multiplicative group Field extensions Contents 1. Introduction...................................................... 406 2. Frobenius modules.................................................. 406 2.1. Preliminaries................................................. 407 2.2. Separability . ................................................ 408 2.3. The Galois group of a Frobenius module.............................. 409 E-mail addresses: [email protected] (J. Morales), [email protected] (A. Sanchez). 1 Research conducted at the 2012 Louisiana State University Research Experience for Undergraduates (REU) site supported by the National Science Foundation REU Grant DMS-0648064. http://dx.doi.org/10.1016/j.jalgebra.2014.06.034 0021-8693/© 2014 Published by Elsevier Inc. 406 J. Morales, A. Sanchez / Journal of Algebra 423 (2015) 405–421 2.4. Integrality................................................... 409 2.5. Description of the splitting field.................................... 411 3. Generic extensions for multiplicative groups ................................ 412 4. Generic polynomials................................................. 415 5. Examples........................................................ 419 References............................................................ 420 1. Introduction An important and classical problem in Galois theory is to describe for a field k and a finite group G all Galois extensions M/L with Galois group G, where L is a field containing k. This can be done by means of a generic polynomial, that is a poly- nomial f(Y ; t1, ..., tm)with coefficients in the function field k(t1, ..., tm)and Galois group G such that every Galois G-extension M/L, with L ⊃ k, is the splitting field of m f(Y ; ξ1, ..., ξm)for a suitable (ξ1, ..., ξm) ∈ L . A related construction is that of generic extensions introduced by Saltman [10]. These are Galois G-extensions of commutative rings S/R, where R = k[t1, ..., tm, 1/d]and d is a nonzero polynomial in k[t1, ..., tm], such that every Galois G-algebra M/L, where L is a field containing k, is of the form M S ⊗ϕ L for a suitable homomorphism of k-algebras ϕ : R → L. Over an infinite ground field k, the existence of generic polynomials is equivalent to the existence of generic extensions as shown by Ledet [8], but the dictionary, at least in the direction {polynomials} →{extensions}, is not straightforward. In this paper, we construct explicitly both a generic extension and a generic polynomial × for groups of the form G = A , where A is a finite-dimensional Fq-algebra and k is an infinite field containing Fq. Both constructions are based on the theory of Frobenius modules as developed by Matzat [9]. An important ingredient is Matzat’s “lower bound” theorem as formulated in [2, Theorem 3.4] that we use to show that the extensions (respectively, polynomials) we construct have the required Galois group. The number of parameters in our construction is not optimal. For example, if 2 A = Mn(Fq), then our method produces a polynomial in n parameters, as opposed to the standard generic polynomial for GLn(Fq)that needs only n parameters [1], [4, Section 1.1]. However, our method has the advantage of being general for all groups × of the form A , where A is any finite-dimensional algebra over Fq. We are indebted to the referee for her/his pertinent and useful comments. 2. Frobenius modules In this section we recall the basic theory and definitions relating to Frobenius modules for convenience of the reader. Most of the material in Sections 2.1–2.3 can be found in [9, Part I], [2]. We include it here for the convenience of the reader. J. Morales, A. Sanchez / Journal of Algebra 423 (2015) 405–421 407 2.1. Preliminaries Let K be a field containing the finite field Fq and let K denote an algebraic closure of K. Definition 1. A Frobenius module over K is a pair (M, ϕ) consisting of a finite-dimensional vector space M over K and an Fq-linear map ϕ : M → M satisfying 1. ϕ(ax) = aqϕ(x)for a ∈ K and x ∈ M. 2 2. The natural extension of ϕ to M ⊗K K → M ⊗K K is injective. The solution space Solϕ(M)of (M, ϕ)is the set of fixed points of ϕ, i.e. Solϕ(M)= x ∈ M ϕ(x)=x , which is clearly an Fq-subspace of M. Let e1, e2, ..., en be a K-basis of M. Clearly ϕ is completely determined by its values on this basis. Write n ϕ(ej)= aijei, i=1 n where aij ∈ K and let A =(aij) ∈ Mn(K). Identifying M with K via the choice of this basis, we have ϕ(X)=AX(q), T (q) q q T where X =(x1, ..., xn) and X =(x1, ..., xn) . Condition (2) of Definition 1 ensures n that A is nonsingular. We shall denote by (K , ϕA)the Frobenius module determined by a matrix A ∈ GLn(K). With the above notation, the solution space Solϕ(M)is identified with the set of solutions in K of the system of polynomial equations AX(q) = X. (1) By the Lang–Steinberg theorem (Theorem 2.5), there is a matrix U =(uij) ∈ GLn(K) such that −1 A = U U (q) , (2) 2 Note that if K is not perfect, the injectivity of ϕ : M → M does not imply condition (2) above. For example, if a ∈ K \ Kq , the map ϕ : K2 → K2 given by ϕ(x, y) =(xq − ayq , 0) is injective over K but not over K. 408 J. Morales, A. Sanchez / Journal of Algebra 423 (2015) 405–421 (q) q −1 where U =(uij). Thus, the change of variables Y = U X over K yields the “trivial” system Y (q) = Y, (3) Fn ⊂ n whose solutions are exactly the vectors in q K . We have proved: ϕ Proposition 2.1. The columns of U form a basis of Sol (M ⊗K K) over Fq. In particular ϕ ⊗ dimFq Sol (M K K)=n. 2.2. Separability n We shall now show that the solutions of (1) are in Ksep. See [9, Theorem 1.1c] for a different argument. Proposition 2.2. Let A ∈ GLn(K) and let x1, x2, ..., xn be indeterminates. Then the K-algebra F = K[X]/ AX(q) − X , T (q) where X =[x1, x2, ..., xn] and AX − X is the ideal generated by the coordinates of AX(q) − X, is étale over K. Proof. Consider the change of variables Y = UX over K, where U is as in (2). Then (q) F ⊗K K = K[Y]/ Y − Y K. 2 Fn q n Corollary 2.3. The solutions of the system of polynomial equations AX(q) = X in K lie n in Ksep. In particular, the matrix U of (2) is in GLn(Ksep). Proof. The solutions of AX(q) = X are exactly the images of X under K-algebra ho- momorphisms F → K. Since F/K is étale, so are all its quotients. This implies that the images of such homomorphisms are contained in Ksep. 2 Definition 2. The splitting field E of (M, ϕ)is the subfield of Ksep generated over K by all the solutions of AX(q) = X. Remark 1. The above definition does not depend on the choice of a basis of M over K. Corollary 2.4. The splitting field E of (M, ϕ) is a finite Galois extension of K generated by the coefficients uij of the matrix U of (2). J. Morales, A. Sanchez / Journal of Algebra 423 (2015) 405–421 409 Proof. The extension E/K is finite, separable by Proposition 2.2. It is normal since a Galois conjugate of a solution X of AX(q) = X is also a solution. Every solution X of (q) AX = X is an Fq-linear combination of the columns of U by Proposition 2.1, thus the coefficients uij of U generate E over K. 2 2.3. The Galois group of a Frobenius module The Lang–Steinberg theorem (see [6, Theorem 1] and [14, Theorem 10.1]) plays an important role in the theory of Frobenius modules. Theorem 2.5 (Lang–Steinberg). Let Γ ⊂ GLn be a closed connected algebraic subgroup − (q) 1 defined over Fq and let A ∈ Γ (K). Then there exists U ∈ Γ (K) such that U(U ) = A. Remark 2. In fact, the element U given in Theorem 2.5 lies in Γ (Ksep)as discussed in Corollary 2.3. Next we state two theorems due to Matzat [9] that play an important role in the determination of the Galois group of a Frobenius module. Theorem 2.6 (“Upper Bound” Theorem). (See [9, Theorem 4.3].) Let Γ ⊂ GLn be a closed connected algebraic subgroup defined over Fq and let A ∈ Γ (K). Let E/K be the n splitting field of the Frobenius module (K , ϕA) defined by A and let U ∈ Γ (E) be an element given by the Lang–Steinberg theorem.